I’ve been playing around with using multiple cameras to capture my classroom this semester. I’m using more than one camera so that I can capture what’s happening in the class as a whole and also what’s happening as I interact with different groups of students. I want to be able to capture video from my point of view for several reasons: (1) I want to record (approximately) what I was seeing and noticing so as to help uncover the signals that led me to decide to intervene with a particular group of students or not; (2) I want to record the kinds of questions that students are asking and how I responded to them–is there something that I could have said that would have helped them more? (3) I want to be able to capture the way students and I talk to each other–what did I position myself relative to them (crouching down, or bending down to talk to them) and did it make any difference in their body language or response to me?
Here’s a sample still from a produced video. You can see four students sitting together in a group of four. (Ideally, you would be able to clearly make out what is on their paper.) In the corner is a wide-angle view of several groups.
Here’s my current setup:
1) Swivl camera base with my Nexus 5X phone as a recording source
2) GoPro Hero4 Silver mounted on my chest with a Sony ECMCS3 microphone.
Good points: both cameras capture at full HD (1920×1080) resolution, and the microphones are by and large capturing important conversations. The video is giving me lots of things to look at.
However, I’ve been plagued by lots of (mostly technical) problems. I have been trying for weeks but I still haven’t found a good method for capturing the video reliably.
Problems encountered so far
GoPro on my chest is mounted in such a way that if I bend down to talk to students, then I just get a shot of the floor or table. I need to constantly remember that I have to position my torso so I get a good view of students’ work and faces.
Swivl camera is pretty horrible because it needs line of sight to track you because it uses IR. It has a remote control that I wear on a lanyard that it uses to track where I move so it should theoretically always have me in the shot, but I’m not in the shot half the time. The only solution I’ve found so far is to stick it up higher above students and have it tilted down slightly.
Audio quality is also pretty bad from the Swivl camera.
The GoPro is supposed to have a feature where I can mark “highlight” moments using a remote control that I wear on my wrist. I haven’t figured out how to export those markers in a useful way. (I experimented with the “video loop” recording mode, but that isn’t really helpful because you only get 5 minute segments.)
Watching the footage from the GoPro makes me nauseous. I am using ffmpeg vidstab to stabilize the video, but it takes hours to process the footage.
Adobe Premiere Pro has a really steep learning curve. If I have to watch another video to figure out how to do something (that I think should be) simple I am going to scream! I’m also behind one version (CS6 instead of CC) and this older version doesn’t have the nifty feature of aligning two clips of video automatically based on the audio tracks.
HD video is great, but the processing time is long, in general. Even moving video files between devices is slow. (Protip: Instead of plugging the GoPro into the computer using a USB cable, take out the SDHC card and stick it in the USB adapter and plug in directly to the computer. Waaay faster. Whoa.. And about 10x faster than that is to put the MicroSDHC card into a SDcard adapter, then put it into a SDcard slot in your computer. I get about 42MB/s transfer rate that way on my computer.)
I am encoding the finished video using H.264. However, if I set the encoding compression too low, the video ends up being huge (15GB). If I set the encoding compression too high, then I can’t make out what is on students’ papers.
Both cameras use exFAT file system, so videos get chopped up into smaller files to avoid 4GB file size limit. That also adds more complications to video processing. On my Nexus 5X, I think it skips a few seconds of video when it transitions from one file to another while recording.
I look pretty ridiculous wearing all this equipment.
This is a work-in-progress post. I hope to have more technical issues sorted out soon.
Since that time, I’ve been thinking about this hierarchy of student needs a lot. I’ve been trying to find scholarly writings on the subject (haven’t found much), and I’ve given several talks featuring these ideas. I’m ready to share these ideas more widely because I think these ideas could be much more fruitful than I originally realized.
A Hierarchy of Student Needs
According to several colleagues who are psychologists, Maslow’s Hierarchy of Needs is a well-respected theory for human motivations. It’s included in most introductory psychology courses. One colleague told me that the theory is so sensible that he finds it difficult to imagine how it might not be true.
Here is how each of category of human needs might translate into students needs in our classes.
Physiological. Thankfully, many of us are blessed to teach in relatively comfortable environments where we are properly sheltered from the environment. Sometimes, the air conditioning makes my classroom too hot or too cold, but by and large I don’t have to worry about my Mudd students’ physiological needs, with the exception that they are often sleep deprived. However, we should remember that far too many students in the United States are food insecure and worry about having a comfortable place to call home.
Safety. Maslow’s original hierarchy had to do with personal and financial safety. While you wouldn’t think that personal safety is an issue that we instructors have to worry about, we must not forget that sexual violence is a big problem at many colleges and universities around the country.
Besides the need for safety from bodily harm, I think there are two other forms of safety to consider in the mathematical classroom: emotionaland intellectual. Lisa’s tweet shows that her student isn’t afraid of being made fun of, being criticized, or being outed to others (except for the students’ mother!). That kind of emotional safety is crucial for students to be open to learning in a classroom. For example, would a student who wears a hijab feel safe in your classroom from ridicule or teasing? Respect for others begins with the instructor. Does the instructor make jokes that single out certain students or groups of students? Does the instructor speak disparagingly about certain groups of people (intentionally or not) or send messages about which groups of students are more or less competent (intentionally or not)? How does the instructor respond to microaggressions and “larger” instances of aggressions, perpetrated by students or the instructor?
All of the protests that have been taking place at colleges and universities over the last few months have underscored the fact that students need emotional safety. It’s disheartening to see how some people are disparaging the need for “safe spaces” by equating them to cocoons where students can never get hurt by disagreements or criticism. The truth is that at many colleges and universities, students of color and other marginalized groups of students go to school in fear of ridicule and scorn. Clearly, that is not an optimal environment in which students can learn.
Intellectual safety is possible when you feel that your ideas are valued by others even if they are incorrect or there is disagreement. Instructors wield a lot of power and respect in the classroom, and that power and respect can be misused. For example, one student shared with me his painful experience asking his professor a question in class. Students were working on a worksheet in groups, and this student had a question that none of his classmates could answer. The student raised his hand, and the professor came over and exclaimed “Oh c’mon!” in condescending and dismissive tone. According to the student, the professor’s response was so loud that other students in the room noticed and a few gasped in shock. The student got into a verbal altercation with the professor about why the professor felt the need to get mad when asked that question. Things did not turn out well. The student was very discouraged and highly unmotivated to learn in class. The student failed the class.
This anecdote reminds me that the way that I respond to incorrect answers and student questions is extremely important. No matter what teachers say to students, the ugly truth is that, yes, students can ask stupid questions. However, it is usually not appropriate to let students know that they have asked a question that they should be able to answer given what they know, especially when students are not confident in their own abilities. I may not respond as severely as the professor in the anecdote above, but I might still send subtle signals to students just from my body language or tone. I’ve learned to develop a poker face that hides my internal reactions to students’ answers or questions. I try to react to all students’ questions and answers in the same way. (That means that when I get a correct answer I also have to modulate my reaction so that I don’t smile approvingly when I get a correct answer and then not smile when I get an incorrect answer.) By the way, this poker-face-approach also has the nice side effect that students don’t know whether their answers are right or not and so I am able to more authentically press them for their justifications.
If you use group work in your classes, intellectual safety is a prerequisite for students to participate. Group work can be scary because it’s an opportunity for students to reveal to each other what they know and understand. I learned this lesson in my course on partial differential equations last semester. On an exit ticket one day, a student wrote this: “Right now, I’m insecure enough about solving problems that the pressure of group work makes me shut down, which only makes it worse.” This student’s lack of intellectual safety prevented him/her from working with others on in-class tasks. Ouch! I felt horrible.
It’s inevitable that students will have different levels of preparation and skill in our classes. What we need to do is to make it clear to our students that there are many different ways to be successful in our classes. It’s not just speed of calculation that makes one better at math. One can be great at observing patterns. One can be great at visualizing functions. One can be great at generalizing ideas. However, if most of our mathematical tasks are procedural and computational, then we risk collapsing mathematical proficiency into a single axis. Valuing all of those different axes is difficult to do in every single task that we assign, but I think it is very possible to make explicit the different ways that students can be successful in mathematics at different points during a semester or unit. (Also, I have written here about how speed and automaticity are often conflated.)
One important thing I’ve also realized since my last posting is that different groups of students can have very different needs in my classroom. It is well-known that men tend have more confidence in their ability to complete tasks based on their own assessment of the necessary skills whereas women tend to doubt themselves even when they have the same skill level. This effect therefore leads to a tendency for women to feel less intellectually safe in a mathematics classroom than men.
Love/Belonging. One could perhaps talk about love in a way that makes sense in the mathematics classroom, but I choose instead of focus on students’ sense of belonging since that seems to me a more straightforward idea. (Please see my previous post about radical inclusivity.) I believe that there at least three different levels of belonging that are relevant here. Students need to feel that they belong to (1) the mathematics classroom, (2) to a larger community of practice of mathematicians, and to (3) any groups that they’ve been assigned to. (The latter will only apply to you if you do group work in your classes.)
I think that there are lots of things that instructors can do to help students feel a sense of belonging. If you care about your students and look after their well-being, you are probably doing things for them that increase their sense of belonging. And, I bet that many of us do these things instinctively without realizing how much they positively affect our students. Here are some examples.
Simply by learning students’ names (and learning how to pronounce them correctly), we give our students a sense that they belong in our class.
A teacher’s sense of humor can sometimes be a great tool for helping students feel a sense of belonging–when you’re in on a joke that only those in your class can understand, that helps you feel like you’re part of a community. This video of a manatee features prominently in all my classes. It’s silly, but it’s also memorable and super effective at building community.
Group work can lead to amazing results, but when implemented poorly it can also lead to disastrous results. When left untreated, status issues in a group of students can lead to students feeling excluded from the group. (Lani Horn has suggestions for addressing status issues here.)
As with emotional/intellectual safety, I have thought a lot about how different groups of students experience different levels of belonging in my classrooms. Any groups of students that are poorly represented in my classroom will naturally feel like they don’t belong to the group as much as the majority students. Therefore, underrepresented students will tend to feel like they belong less than majority students. Underrepresented students will probably also feel less belonging to the mathematics community as a whole because they don’t see many people like them getting tenure, attending conferences, winning professional awards, publishing research.
The messages that we send to our students about belonging are subtle but important. Sometimes we have the right intentions but we do things that make students feel like they don’t belong. For example, Lauren Aguilar, Greg Walton and Carl Weiman point out that if you continually tell a student that she can succeed and you don’t tell that to other students, that student might begin to wonder whether you doubt her ability to succeed. Or if you set up additional office hours and make a special effort to invite women and minorities to attend, you might be sending the message that you think all women and minority students have less adequate preparation.
Esteem. The Wikipedia page on Maslow’s Hierarchy of Needs describes esteem as “a need to feel respected… to have self-esteem and self-respect.” The analog of esteem in the mathematics classroom is a student’s self-concept as a learner of mathematics. Every time a student is presented with a mathematical task, that student’s self-concept is activated in the form of an appraisal of her/his own abilities as a learner of mathematics based on prior achievements, comparisons with peers’ abilities, and perceptions of the mathematical task at hand. That appraisal of success at the task gives the student confidence or reluctance to take on the task. When I taught high school, I noticed that students with low self-concept would become disruptive or disengaged when presented with a task that they thought they would not be able to complete. These behaviors, unconsciously or consciously, help the student avoid the possibility of failure and public or private shaming from the teacher so as to preserve his/her self-esteem. (I found this old blog post from 2009 that gives a specific example of this.) At Harvey Mudd, instead of disengaging or becoming disruptive, students with low-concept will procrastinate on their work and rationalize their poor performance as due to lack of time devoted to the task.
It is important to note that self-concept correlates very strongly with student performance, and may be one of the variables that is most strongly correlated to student performance. (See John Hattie’s book Visible Learning.) One of the most important ways that we can attend to students’ self-concept is by using formative assessment to have a very detailed understanding of students’ skills and understanding, and to give them tasks that are appropriate to their level of mathematical development.
Self-actualization. Ok, if you’re like me, you probably approach this word with a little hesitation about sounding self-helpy…. But, the according to Maslow himself, self-actualization refers to the desire to accomplish everything that one can, to become the most that one can be. That makes a lot of sense to me. When one has achieved a certain level of success with mathematics I think it is natural to wonder what else one can achieve and then to try to do it. That need to test one’s boundaries is a wonderful human characteristic.
Maslow theorized that the four foundational needs (physiological, safety, love/belonging, esteem) are “deficiency” or “basic” (see Fig 1) in the sense that those needs must be met before an individual will strongly desire any higher level needs. In other words, if I do not have enough food or water to survive, I am going to spend most of my energy making sure that I can meet those needs first before I think about my physical safety and love/belonging. People who don’t have these deficiency needs met will feel anxious. I imagine that the same is true about students’ needs in the classroom. If a student doesn’t feel intellectually safe, it’s a safe bet that the student will feel anxious and that anxiety will weigh on the students’ ability to learn.
How the Hierarchy of Student Needs Relates to Equity and Inclusion
Since my last post, I’ve thought a lot about how this hierarchy of student needs relates to equity and inclusion. The key insight that I come to over and over again is how different groups of students in my classes have different levels of needs. Majority students tend to come to my classes with a more secure sense of intellectual and emotional safety and sense of belonging to the classroom. Perhaps that is one reason why we often find differences in educational outcomes when we aggregate students into different groups.
“In addition to providing evidence that active learning can improve undergraduate STEM education, the results reported here have important implications for future research. The studies we metaanalyzed represent the first-generation of work on undergraduate STEM education, where researchers contrasted a diverse array of active learning approaches and intensities with traditional lecturing. Given our results, it is reasonable to raise concerns about the continued use of traditional lecturing as a control in future experiments … The data suggest that STEM instructors may begin to question the continued use of traditional lecturing in everyday practice, especially in light of recent work indicating that active learning confers disproportionate benefits for STEM students from disadvantaged backgrounds and for female students in male-dominated fields.”
A general consensus is building that many active learning strategies improve learning outcomes for all students, but that they also improve learning outcomes disproportionately for women and underrepresented students. Here are three examples of documented cases of exactly that:
(Each of these articles is worthwhile to read. I would appreciate it if others can point out additional examples of this kind of research.)
Of course, these studies are empirical. They observe that these disproportionately positive effects occur for disenfranchised or marginalized groups of students, but they can’t explain why that happens. Could this hierarchy of student needs be that explanation?
Suppose you have a teacher who implements group work effectively in a mathematics classroom. In this scenario, imagine what happens to a woman who initially comes into the class doubting her abilities. During the course of doing mathematics with other students in class, this woman realizes that others are having the same struggles too, or that she’s actually more capable than she realized. That realization increases her sense of intellectual safety, sense of belonging to the class, and self-concept as a learner of mathematics. The same thing could happen if an instructor used clickers/plickers/etc in class along with questions that generate meaningful dialogue and surface common misconceptions.
Suppose you have a teacher who is really great at orchestrating classroom discourse. Imagine an African-American student in this teacher’s class, who has received lots of signals that previous teachers have doubted his mathematical ability. This teacher is great at making sure every students’ idea is taken seriously and is worthy of consideration. One day, the student tosses out an idea that some students initially dismiss, but the teacher carries it out to its logical conclusion and finds it to be innovative and correct. The student’s self-concept as a learner of mathematics increases as he begins to realize that perhaps he’s skilled in mathematics in a way that he and others have never appreciated, and he begins to call into question all of the previous signals he’s received from others. Other students take notice of his abilities too, and that increases his sense of belonging in the class.
I’m sure you could come up with similar scenarios too.
Once we accept that there are certain teaching practices (for example, active learning strategies) that happen to be very effective also happen to promote greater equity and inclusion, we arrive at this question: Is inclusive teaching the same as effective teaching? I believe that this statement is true, but only in part.
Inclusive teaching is a set of principles, goals, and practices, grounded in research, experience, and commitments to social justice. A large subset of these principles, goals, and practices could easily also be described as effective teaching. And in fact, it may be difficult to distinguish one from the other simply by looking at a sample of teaching practices. (I wrote more about this here.)
Inclusive teaching adds to effective teaching a framework for understanding why teaching is effective, along with an intentionality of producing more equitable outcomes for students. A faculty member may teach effectively without consciously considering inclusiveness, but by being more intentional about the desired outcomes of learning and designing every aspect of the learning to address students’ needs, they could help to create even better results.
These ideas seem so natural to me and yet I feel like I’ve just scratched the surface. There is more to uncover and think about, I’m sure. For example, if this hierarchy of students needs can help to explain why different teaching strategies lead to different results for different groups of students, then perhaps researchers should measure students’ sense of safety, belonging, and self-concept along with their learning outcomes when they compare different interventions.
I haven’t been posting much because of the busy-ness of last semester. Now that grades have been submitted, I’ve been reflecting on the partial differential equations (PDEs) course that I taught. (All previous posts about this class can be found here.)
I firmly believe that we must evaluate our own teaching if we want to improve, and that one of the best ways to gather data on our teaching is to ask our students. Students aren’t always the best judge of how much they have learned, but I trust my Mudd students’ ability to tell me about their experiences and opinions about the course. Here is what students said in their comments on an end-of-semester evaluation survey.
Comments about students’ perceptions of the course and their overall experiences:
I really appreciate your comments at the beginning of class that you realized that most of the applications you were planning on presenting were thought up by dead white guys and that that might cause some people dismay. I think that recognizing that that’s a problem in math/science that permeates into classrooms is important, and you saying that out loud helped me feel more like I belonged in the classroom even if I am not a white male.
This class somehow made me enjoy solving PDEs even though the past three DE classes I’ve taken convinced me I just really hate DEs. Nice job.
I started solving random PDEs in my free time, from which I deduce the class was pretty interesting.
I also want to thank you for being such a dedicated teacher. Your lectures and notes in numerical analysis and PDEs were effective in delivering material and it never felt like there was anything “hidden” about the subjects that I couldn’t figure out without some closer reading. I also think that the structure of this class was awesome, because it encourages you to learn all aspects of the subject, even if you missed that specific part of the semester.
I feel that [Prof. Yong] is extremely understanding and is very approachable to students, regardless of how comfortable they are with the material.
Overall, I noticed that student engagement was high. I was really pleased that I was able to change some students’ opinions of differential equations.
On the end-of-course survey, I asked students to indicate their affinity for the following statements on a scale from 1 (strongly disagree) to 5 (strongly agree). (A total of 32 students responded to the survey, though not every student responded to every question.)
“The students and instructor for this course created a welcoming community of learners.” Average response: 4.59 / 5
“In this class, I was able to express myself (whether it was to answer a question, or to say that I didn’t know how to do something) openly without judgment or ridicule from my instructor.” Average response: 4.81 / 5
“I generally felt secure and confident to speak in this class (to answer a question or ask a question or something else) when I wanted to.” Average response: 4.44
“I feel secure and confident to speak in my classes at Mudd in general.” Average response: 4.13 / 5
“I feel like an outsider in this class.” Average response: 1.68 / 5
“The instructor was respectful to all students in the class.” Average response: 6.91 / 7
Student comments about the proficiency assessment system:
Since the proficiency assessment (PA) system was a big change for me and students, naturally there were lots of comments about this part of the course. I tallied up all of the types of comments that I received about the system: 16 favorable comments, 4 negative comments, and 1 mixed comment.
Here’s a selection of the positive feedback:
The proficiency assessments are spot on in encouraging learning–I feel like I’ve learned from them while satisfactorily representing my understanding.
I’ve never had a class where I could schedule tests like this one, but wish I had! I have gotten a lot out of this freedom to prepare when I have time, and to take more advanced versions once ready for them.
Also, there seems to be the philosophy that students deserve credits when he/she knows how to solve the problems and not only when he/she can solve the problems within the exam time. I feel like this grading philosophy is more applicable to the work in real life.
Since I could take it many times, I didn’t care much about getting the right answer. Instead I was able to focus more on improving myself each time I take the PA.
I like how the proficiency assessments encourage me to understand the material at my own pace in a less stressful way.
There were several classes that I’ve taken at Mudd where I didn’t learn the material by the time the exam came, and so there was no reason for me to learn it afterwards. I also like the ways in which [the PA system] discouraged cheating: since you have the opportunity to retake exams, there is less of an incentive to be dishonest about it. For me it was also great because it meant that it was never too late for me to try to catch up. One of the most depressing situations that I encountered at Mudd was having several exams during one week and feeling like I could have performed better if the schedule had just worked out differently. I personally think that your curriculum is the right direction for a lot of the classes at Mudd, so thanks for being willing to try something new out.
On the whole, I think I was able to meet my objective of coming up with a system to assess students’ understanding but in a way that gave students more agency and flexibility, and that promoted students’ growth mindset about learning PDEs.
On the end-of-course survey, I asked students to indicate their affinity for the following statements on a scale from 1 (strongly disagree) to 5 (strongly agree).
“The proficiency assessments fairly assessed my understanding of PDEs.” Average response: 4.56.
“I was able to demonstrate my understanding of the course material through the proficiency assessments and final exam.” Average response: 4.50
I will definitely use this system again in the future, but I need to figure out how to reduce the impact on my time and on students’ time. Also, I need to think more carefully about how students’ grades are calculated based on their assessment scores.
I ended up writing four sets of proficiency assessments, on four different subtopics. There were standard and advanced assessments for each subtopic. The maximum score attainable on a standard assessment was 23/25, and the maximum for an advanced assessment was 25/25. The standard assessments contained tasks that I would have used as final exam questions, and advanced assessments contained more challenging tasks that involved novel situations or challenges that they had not encountered, but that they could if they dug deeper into the course content. My rationale for this arrangement is that a standard level of attainment would correspond to a 92%, which in my mind is an A-. To get an A in a course, I feel that some effort above and beyond the normal set of expectations is required. I required students to reach the standard level of attainment before attempting a more advanced PA. This had the effect of requiring students to take at least eight separate assessments, if they wanted to get an advanced level of attainment for all four subtopics.
I’m really mixed about the PA format. On the one hand, I think I’ll probably have learned the material better than usual by the end of this course, simply because I’ll have had more tests on the subject scattered throughout the semester. However, I also feel like the PAs took up so much time (both for the student and the instructor grading them) that they caused more stress than a midterm would have.
I feel that the course had too many disparate requirements. The presence of PAs, homeworks, a final test, and a final project made it an overwhelming experience. I would advocate for a more efficient PA system that doesn’t require as much time outside of class, since the current PA system feels more like it’s testing students for how much of their free time they can sacrifice as opposed to their actual knowledge.
I liked the concept behind PAs, but found it somewhat annoying that in order to get full credit on all of the PAs you would have to schedule 8 different time slots. One idea I had to ease up on this sheer time commitment would be to have on each PA the option to choose between the 23 point problem or the 25 point problem.
I don’t feel that the PA system ended up fairly assessing my understanding. I feel that they got close to providing a good assessment, but with the time restrictions I’ve faced this semester, I was unable to take the advanced PAs. I had a very busy schedule this semester, and I was sick for multiple weeks in the middle of the semester, so I didn’t get to take my last regular PA until the end of finals week.
I got perfect scores on the 3 PAs I’ve gotten back so far, and I suspect I also did perfectly on the last PA I took this morning. But I just didn’t have the time to attempt the advanced PAs, especially given that my score isn’t even guaranteed to increase, so it was very hard to justify adding something else into my already exhausting schedule. I feel that I have a very strong understanding of the material, but my time restrictions only allowed me to demonstrate a “basic” understanding.
Personally, I feel that scheduling time for four 90-minute assessments isn’t a huge burden on students, but some felt that way. One thing I will try in the future is to set aside some class time for students to take the proficiency assessments, so they aren’t required to use out-of-class time.
Overall, I’m really pleased with how the course went, even though it was my first time teaching the course and I was experimenting with lots of new ideas. I was trying to attend to students’ sense of belonging to the class all semester, and I think I was successful at that.
I consider the proficiency assessment experiment to be a strong success. I want to continue to refine and improve it. One important side effect of the proficiency assessment system is that I got to know all of my students much better than I normally would have. Another side effect is that the system enabled some students who would probably failed the course otherwise to pass and do well. For example, one student who had some family issues and was absent for almost 2/3 of the class was able to finish the course on time.
Proficiency assessments (PAs) are proceeding nicely in my partial differential equations (PDEs) course. I’m posting some of the logistical details here in case it helps other instructors. PAs are my attempt to allow students more flexibility in demonstrating their mathematical proficiency in my class. I wrote more about my intentions behind these PAs in this post.
First of all, I’ve reduced the number of PAs from 5 to 4. Here are the four PAs for this course:
Be able to solve a first-order PDE using the method of characteristics, in both linear and nonlinear cases. (Nonlinear PDEs may involve shocks.)
Be able to use the separation of variables technique to solve a homogeneous PDE problem.
Be able to solve an inhomogeneous PDE problem using an appropriate change of variables, or the eigenfunction expansion method.
Be able to use integral transforms (Fourier and Laplace) to solve PDE problems involving infinite or semi-infinite domains, and be able to identify how general solutions are convolutions involving Green’s functions.
The initial list of PAs was created while I was still constructing the class and once I got deeper into things I realized that one of the things that I listed on my previous list was better assessed on the final examination.
Scheduling 40 students to take these assessments has been a bit of a challenge, but I think we’ve solved that problem. Each PA is supposed to be completed during a 90-minute session without notes, calculators or outside assistance. But, students can use more than 90 minutes. That time is just a suggestion for the length of time to block out in their schedules.
I’m not terribly worried about cheating because of the strong honor code here at Harvey Mudd, but I want to be careful since the assessments can be taken at any time and I don’t want copies of them floating around. So, I needed to find a way for students to make appointments to take a PA. With the help of the students, I came up with this mechanism for taking PAs. There are four different ways to schedule an appointment:
Students can email me to set up an appointment. I have a calendar online so students can see when I have open blocks of time when they can sign up for an appointment.
Students can also contact our department administrative aide to make an appointment during business hours (Mon-Fri).
I have two graders/tutors for this course and they have additional tutoring time on Wednesday evenings. Students can arrange to take a PA under their supervision.
Finally, a group of students can get together and arrange to take an assessment together, at any time, even nights and weekends. These “group-proctored” sessions have turned out to be the most popular so far. I make arrangements to leave the assessments in some secure place, then they put the completed assessments in an envelope and slide it under my door if I am not at work at that time.
Some of you might be skeptical about students taking an exam/quiz on their own without supervision. it really does work here at Mudd, though. When students want to set up a group-proctored session, I also email the entire class to let them know about the opportunity. That gets groups of students together who aren’t in the same friend groups, so there is a bit more accountability.
So far, I’ve been meeting with each student after every PA to talk briefly about her/his performance. It is taking a lot of my time, but I really like having that personal connection with every student. I have far more information on how well each student is mastering the material than I have in previous classes.
Finally, I’m also really enjoying starting every class with 2-3 minutes on some cool application of PDEs: traffic simulation, tsunamis, bread baking, digital image restoration are some of the more interesting ones so far, in addition to the usual diffusion, advection/convection, Laplace problems. Students seem to really enjoy it too. If you have have more cool applications of PDEs to share, please let me know!
A student missed class because he was sick. He emailed me to let me know and I asked him to come by my office to catch up on things. When he came to my office for his appointment he was visibly nervous: uncomfortable body language, never made eye contact with me, spoke erratically.
After about 30 minutes, he told me that he’s never gone to a professor’s office hours before. I thought to myself, WHAT??? YOU’RE A JUNIOR MATH MAJOR! HOW IS THIS POSSIBLE????!!! I didn’t say that, of course. We just kept chatting and I encouraged him to email me with questions and to stop by again. Thankfully, he listened to my encouragement: we’ve been exchanging emails like rapid-fire IMs the last few days.
Perhaps this kind of thing is not unusual at larger institutions, but Harvey Mudd College is a small school with just 820 or so students. Faculty and students get to know each other well and it is a very close-knit community. In the 14 years I’ve taught here, I’ve not heard of a third-year student who has never gone to a professor’s office to ask for something. So on one hand it seemed to me a really unusual thing to happen.
But on further reflection, I started wondering if maybe it was more common than I realized. In every class I teach, I estimate that 1/3 of the students come to my office regularly, 1/3 come occasionally, and 1/3 never come at all. I suspect that latter 1/3 group is mostly made up of students who never visit a professor’s office.
This interaction made me think about the “hidden curriculum” of higher education–those unspoken things about college that some of our students (mostly those with a lot of privileges) know, and that others don’t know. For example, one colleague shared about how she went to college and thought it was strange that there were so many people named “Dean” until she realized that “Dean Smith” meant “Dean of Something-Or-Other whose last name is Smith”, not that the person’s first name was Dean. The fact that students are supposed to go to office hours to interact with their professors is another thing that many students don’t realize. Some students think that going to ask for help is a sign of trouble or weakness, whereas other students have figured out that going to talk to a professor is a normal and routine part of going to college.
So to teach inclusively means that this welcome spirit must also extend to my interactions with students outside of the classroom. Am I making it clear to all students that they are welcome–no, expected–to come to my office to chat with me? If I don’t do that, then I am, deliberately or not, favoring some students who know about the “hidden curriculum” over those that don’t.
Together with Pam Mason, I help to design professional development for Math for America Los Angeles. Right now we have about 85 Teaching Fellows and Master Teaching Fellows in our program.
This year, we decided to implement something new to help “make thinking visible”–that’s my best version of what to call this thing. It’s not new, just new for us. But, I think it’s important enough to write about, hence this post.
Here’s what I mean by “making thinking visible.”
We teachers use phrases like “Say your ‘because’s” as a signal to students and each other that we care about student thinking. We want our students to understand mathematics, not just calculate, though that is important too. So when a student talks about mathematics in our class, we always press them to justify their answers. They need to say their “because”s. For example, we want students to say things like “I think the next step should be to square both sides of the equation because we are trying to find an equivalent equation without square roots.”
Well, the same thing should be true for us teachers too. When we talk to each other as professionals, we should also be making our reasoning and thinking more visible to each other so that we can learn from one another.
At least in Math for America Los Angeles, we haven’t been so good at that. We have been really good at sharing things with each other. We share our lessons, we share our resources, we share our struggles.
But when we say “I like this lesson”, do we explain why to each other?
When we say “I think students struggle here”, do we explain why?
When we say “This unit is going to take 2 more days than we expected”, do we explain why?
Some of the conversations behind these statements are actually the conversations we need to have with each other. They are meaningful and juicy because they reveal our beliefs, logic systems, and understanding about teaching and learning. If we want to really share our knowledge with each other, we should be sharing about these things rather than just sharing lessons and ideas.
Put another way, if we simply shared lessons with each other, that would be equivalent to students just telling us answers without justification. They might be arriving at the right answers for the wrong reasons! Of course we don’t want that to happen, with our students and shouldn’t let that happen for each other either.
I think making our thinking visible is not a complicated thing to do. We don’t need special training to do this, since we teachers are already in the habit of asking good questions and pressing for understanding with our students. We just have to be vigilant to do that we each other because we value each other as professionals in a craft that is deep and worthy of study.